. N T ] 2 2 Ju n 20 06 REPRESENTING PRIMES AS x 2 + 5 y 2 : AN INDUCTIVE PROOF THAT EULER MISSED

We present an elementary inductive proof which Euler could have obtained, for the corresponding result as the title indicates, had he refined a bit his proof for Fermat’s assertion on representing primes as two squares. It is really a pleasure to start with the story told by Cox in the nice book [2]. All letters in this note stand for nonnegative integers, unless otherwise specified. 1. Fermat (Pierre de Fermat, 1601–1665), who did important work on representing primes as x + ny, stated, but did not write down a proof, that he had proved by his favorite infinite descent method the following: (i) Every prime p ≡ 1 mod 4 is a sum x + y. (ii) Every prime p ≡ 1, 3 mod 8 is a sum x + 2y. (iii) Every prime p ≡ 1 mod 3 is a sum x + 3y. He also conjectured but could not prove that (iv) The product of two primes, each of which is ≡ 3, 7 mod 20, is a sum x+5y. 2. Euler (Leonhard Euler, 1707–1783) heard of Fermat’s results and spent 40 years in proving (i)–(iii) and considering their generalizations, which finally led him to the discovery of the quadratic reciprocity. He experimented on many examples and discovered more. Some of his conjectures which he could not prove are: (v) Every prime p ≡ 1, 9 mod 20 is a sum x + 5y. (vi) For every prime p ≡ 3, 7 mod 20, 2p is a sum x + 5y. (vii) A prime p = x + 27y if and only if p ≡ 1 mod 3 and 2 is a cubic residue modulo p. (viii) A prime p = x + 64y if and only if p ≡ 1 mod 4 and 2 is a biquadratic residue modulo p. 3. Lagrange (Joseph-Louis Lagrange, 1736–1813) and Legendre (Adrien-Marie Legendre, 1752–1833) later developed the form theory as well as the genus theory to prove (iv)–(vi). In deed, they could prove (v) and that (v) Every prime p ≡ 3, 7 mod 20 is a sum 2x + 2xy + 3y (where one of x, y may be negative). Then (iv) and (vi) follow immediately from the following two identities: (2x + 2xy + 3y)(2a + 2ab+ 3b)= (2ax+ bx+ ay + 3by) + 5(bx− ay); 2(2x + 2xy + 3y)= (2x+ y) + 5y. Date: June 21, 2006. The author is supported by a CNPq-TWAS postdoctoral fellowship while writing this paper.